Twitching of eyes have great significance in our scriptures. There are various description of auspicious and inauspicious omens in our scriptures and eye twitching is one of them.
Twitching of Sita's left body parts is mentioned in Valmiki Ramayana Sundara Kanda. In chapter 27 verses 48 to 50.
निमित्तभूतमेतत्तु श्रोतुमस्या महत्प्रियम् || ५-२७-४८
दृश्यते च स्फुरच्चक्षुः पद्मपत्रमिवायतम् |
Being a cause to listen to a great pleasure of Her, this throbbing eye long like a lotus petal is being seen my me.
ईषच्च हृषितो वास्या दक्षिणाया ह्यदक्षिणः || ५-२७-४९
अकस्मादेव वैदेह्या बहुरेकः प्रकम्पते |
One left arm of this straight forward Seetha suddenly has become erect and is slightly trembling."
करेणुहस्तप्रतिमः सव्यश्चोरुरनुत्तमः || ५-२७-५०
वेपमानः सूचयति राघवं पुरतः स्थितम् |
Seetha's best left thigh equal to an elephant's trunk is trembling indicating Rama's standing in front of Her.
Valmiki Ramayana Yuddha Kanda 6.4.7 mentions it where Lord Ram tells about his twitching of eyelid.
उपरिष्टाद्द् हि नयनम् स्फुरमाणम् इदम् मम ||
विजयम् समनुप्राप्तम् शंसति इव मनो रथम् ।।
"My eye which is twitching on the upper lid, is proclaiming as it were, my desire of victory coming nearer."
Valmiki Ramayana Yuddha Kanda 6.65.50 mentions bad omen signalling Kumbhakarna
निष्पपात च गृध्रेऽस्य शूले वै पथि गच्छतः |
प्रास्फुरन्नयनम् चास्य सव्यो बाहुरकम्पत ||
A vulture descended on the spike of Kumbhakarna, as he was sallying forth along the road. &His left eye twitched and his left arm throbbed.
Similarly Ramcharitmanas also mentions about twitching of eye.
Sundar kanda Ramcharitmanas Chaupai of 34th Doha
राम कृपा बल पाइ कपिंदा ।
भए पच्छजुत मनहुं गिरिंदा।।
हरषि राम तब कीन्ह पयाना ।
सगुन भए सुंदर सुभ नाना।।
जासु सकल मंगलमय कीती ।
तासु पयान सगुन यह नीती।।
प्रभु पयान जाना बैदेही ।
फरकि बाम अंग जनु कहि देहीं।।
जोइ जोइ सगुन जानकिहि होई ।
असगुन भयउ रावनहिं सोईं।।
Getting more power from His grace the monkey chiefs vied as it were with huge mountains equipped with wings. Råma then sallied forth glad of heart
and many were the delightful and auspicious omens that occurred to Him . It was in the fitness of things that good omens should appear at the time of His departure (on an expedition) whose glory itself embodies all blessings. Videhas Daughter came to know of the Lords march; the throbbing of Her left limbs apprised Her of the same as it were. But what were good omens for Janakas Daughter foreboded evil for Ravana.
Also, In the chaupai of 99th Doha of Ramcharitmanas
With many such words did Trijatå comfort Sita and then returned to her residence.
As She recalled Ramas kind disposition Videhas Daughter was overwhelmed with the anguish of separation from Him. She reproached the night and the moon in many ways. "The night has already assumed the length of an age and does not end.", She added. Disconsolate at Her separation from Lord Råma, Janakas Daughter grievously lamented within Herself. When Her agony of separation grew acute, Her left eye and arm throbbed. Considering it to be a good omen, She took heart and said to Herself, The gracious Hero of Raghus line will surely meet me.
Hence, twitching of right part of male and left part of female [especially eyes] is considered auspicious and vice versa.
Omen signs are greatly discussed in our scriptures. You can't tell them as mere superstition. Omens are studied under 'Shakun Shastra'.
Every scripture mentions of omen before some important omens. For eg. Mahabharat mentions crying of jackals and other inauspicious signs when Duryodhan was born. Inauspicious omens were seen just after Shishupals birth and his parents were just about abandoning him. Similar is the case in the birth of Kamsa. Numerous inauspicious omens are described in great detail by Veda Vyas before the Mahabharat war. Similarly Ram mentions about colour of sky and stars before Ramayan war and so on many...
Hence seeing omens are a part of our scripture and culture. Hence, We should try to understand the meaning of omen we see.