Are there any famous patriotic/nationalist hymns/verses about Bharatavarsha or jusr one's own country found in the Vedas, Itihasas or any Hindu scriptures?
Yes, there are several patriotic hymns Hindu Scriptures.
Yajurveda 22.22 states:
आ ब्रह्मन् ब्राह्मणो ब्रह्मवर्चसी जायतामा राष्ट्रे राजन्यः शूर इषव्योऽअतिव्याधी महारथो जायतां दोग्ध्री धेनुर्वोढानड्वानाशुः सप्तिः पुरन्धिर्योषा जिष्णू रथेष्ठाः सभेयो युवास्य यजमानस्य वीरो जायतां निकामे निकामे नः पर्जन्यो वर्षतु फलवत्यो न ओषधयः पच्यन्तां योगक्षेमो नः कल्पताम् ॥
English Transliteration: A brahman brAhmaNo brahmavarchasI jAyatAmA rAShTre rAjanyaH shUra iShavyo&ativyAdhI mahAratho jAyatAM dogdhrI dhenurvoDhAnaDvAnAshuH saptiH purandhiryoShA jiShNU ratheShThAH sabheyo yuvAsya yajamAnasya vIro jAyatAM nikAme nikAme naH parjanyo varShatu phalavatyo na oShadhayaH pachyantAM yojakShemo naH kalpatAm ||
English Translation: O the Supreme Lord of the world, in this country of ours, let the Brahmins rise as a community of brilliant men of divinity and illustrious scholars of universal knowledge. Let the Kshatriyas rise as a force of heroic warriors of the chariot and the routers of the enemy with unfailing arrows. May the cows be fertile and abundant in milk, the oxen, carriers of heavy load and coursers be swift; the woman, kind and generous; and may the children (people) of the yajamana; the ruler, be brave and fearless, the victorious riders of the chariot, youthful and civil members of the government. May the clouds rain in showers for us according to the need of the season. May the herbs and trees mature and ripen with abundant fruit. And maybe this land be securely blest with a ceaseless cycle of growth and protection of happiness and prosperity.
Srimad Bhagavatam 5.19.21 states:
एतदेव हि देवा गायन्ति— अहो अमीषां किमकारि शोभनं प्रसन्न एषां स्विदुत स्वयं हरि: । यैर्जन्म लब्धं नृषु भारताजिरे मुकुन्दसेवौपयिकं स्पृहा हि न: ॥ २१ ॥
English Transliteration: etad eva hi devā gāyanti — aho amīṣāṁ kim akāri śobhanaṁ prasanna eṣāṁ svid uta svayaṁ hariḥ yair janma labdhaṁ nṛṣu bhāratājire mukunda-sevaupayikaṁ spṛhā hi naḥ
The Devas said: How wonderful it is for these human beings to have been born in the land of Bharatavarsha. What pious acts of austerity they may have executed the past, for Hari himself is pleased with them. In all humans, those who are born in Bharatvarsha, are so fortunate that they have already executed themselves in devotional service of Mukunda. Therefore, we the Devas, always aspire to achieve human births in Bharatvarsha.
The Vishnu Purana Canto 2, Chapter 3 states:
In Jambu-dwípa, Vishnu, consisting of sacrifice, is worshipped, as the male of sacrificial rites, with sacrificial ceremonies: he is adored under other forms elsewhere. Bhárata is therefore the best of the divisions of Jambu-dwípa, because it is the land of works: the others are places of enjoyment alone. It is only after many thousand births, and the aggregation of much merit, that living beings are sometimes born in Bhárata as men. The Devas themselves exclaim, "Happy are those who are born, even from the condition of Devas, as men in Bhárata-varsha, as that is the way to the pleasures of Paradise, or the greater blessing of final liberation. Happy are they who, consigning all the unheeded rewards of their acts to the supreme and eternal Vishńu, obtain existence in that land of works, as their path to him. We know not, when the acts that have obtained us Swarga shall have been fully recompensed, where we shall renew corporeal confinement; but we know that those men are fortunate who are born with perfect faculties in Bhárata-varsha. From this region Swarga is obtained, or even, in some cases, liberation from existence; or men pass from hence into the condition of brutes, or fall into hell. Swarga, emancipation, a state in mid-air, or in the subterraneous realms, succeeds to existence here, and the world of acts is not the title of any other portion of the universe."
In the Valmiki Ramayana 6.124.17B, Sri Rama says:
जननी नन्म भूमिश्च स्वर्गादपि गरीयसी
English Transliteration: jananī janmabhūmiśca svargādapi garīyasī
English Translation: Mother and motherland are far superior to even Swargaloka.
By the way, this verse is also incidentally the national motto of Nepal.
Here is one.
It seems that the gods sing songs to this purport. Those who live in this part of the world called Bharata are in fact blessed. In order to enjoy the fruits of heaven and salvation they are born as human beings, casting off their deity-hood. (Garuda Purana, Dharma Khanda, Chapter I)
Yes, there are several patriotic/nationalist hymns found in Hindu scriptures. For example, here are some Atharva Veda Suktas which are centered towards peace, stability, unity, prosperity of nation and its people. Atharva Veda is full of such patriotic/nationalist hymns. Below are some of the Mantras from Atharva Veda.
Atharva-Veda Kanda 3 - Sukta 8 -राष्ट्र धारण सूक्त.- A charm to secure the submission, love, and fidelity of kinsmen.
This is a patriotic hymn for unification of nation . It's said in the mantra that lets our hearts be one.
अहं गृणभामि मनसा मनांसि मम चित्तमनु चित्तेभिरेत |
मम वशेषु ह्यदयानि व: कृणोमि मम यातमनुवत्र्मान एत || AV 3.8.6||
I with my spirit seize and hold your spirits. Follow with thought and wish my thoughts and wishes. I make your hearts the thralls of my dominion; on me attendant come thy way I guide you.
This mantra is for stability of a Nation.
आ यातु मित्र ऋतुभि: कल्पमान: संवेशयन् पृथिविमुस्त्रियाभि : |
अथास्मभ्यं वरुणो वायुरग्निर्बुहद् राष्ट्रं संवेश्यं दधातु || AV 3.8.1||
Let Mitra come, arranging, with the Seasons, lulling the Earth to rest with gleams of splendour And so let Agni, Varuna, and Vāyu make our dominion tran-quil and exalted.
Here is an English Translation by Ralph T.H. Griffith
Atharva Veda - Kanda 1 - Sukta 29 -Rashta Abhivardhan Sukta - राष्ट्र -अभिवर्धन सूक्त -
This Sukta is for Development of workforce of people for the Prosperity and welfare of Nation.
अभिवर्तेन मणिना येनेन्द्रो अभिवावृधे | तेनास्मान ब्रह्मणस्पतेsभि राष्ट्राय वर्धय || AV 1.29.1 ||
O Brahmanspate ! Develope us for the welfare of the nation through the gem by which Indra flourished.
Atharva Veda - Kanda 6 - Sukta 42 -ParasparaChittaEkikarn Sukta - परस्परचितैकीकरण सूक्त -- Mutual Unity of hearts.
This Sukta is for mutual unity among people of the nation so they can work unitedly as a team by uniting their minds i.e. uniting their minds as one single mind and working towards prosperity welfare and development of nation. The concept integrity among people of a nation is visible in this Sukta which is a good example of patriotic hymns.
सखायाविव सचावहा अव मन्युं तनोमि ते |
अधस्ते अशम्नो मन्युमुपास्यामसि यो गुरु : || AV 6.42.2 ||
By uniting all our minds together , Let us work together with one mind. We discard our anger towards each other ,we crush our anger towards each other with heavy stone.
together let us walk as friends: thy wrathful feeling I remove. Beneath a heavy stone we cast thy wrath away and bury it.
English Translation of the above mantra.