Some advaitins claim that Nirguna Brahman spoke through Krishna, but is this in accordance with advaitin theory that Brahman is without any characteristic, is speaking not considered a characteristic, or an act of an independent individual? According to advaita.
According to Advaita, there is only Brahman, and nothing more. Everything you see, the world, is just Brahman.
Vivekacudamani, Sri Adhishankaracharya - verse 231
brahm'aiv'edaṁ viśvam ity'eva vāṇī śrautī brūte'atharva-niṣṭhā variṣṭhā tasmad etad brahma-mātraṁ hi viśvaṁ n'ādhiṣṭhānād bhinnatā'āropitasya
The scriture [atharva veda] says this world is Brahman. All is Brahman. Therefore, all this world is nothing but Brahman. What you impute to a thing [superimpose] is not separate from that to which you impute [the substratum]. The basis of all is Brahman. Therefore, what you call the world is not the world, but Brahman.
So, Sri Krishna and everyone else are Brahman. When Sri Krishna speaks is only Brahman who speaks. But also when a little ant die, it is just Brahman who die.
yac cā'pi sarvabhūtānāṁ bījaṁ tad aham arjuna na tad asti vinā yat syān mayā bhūtaṁ carācaram
And further, whatsoever is the seed of all existences that am I, O Arjuna; nor is there anything, moving or unmoving that can exist without Me.
When you see Brahman as a little insect with 6 legs, etc, you call it "ant Brahman" (usually without the Brahman part, just ant ;) ). When you can put qualities to something, that is saguna. And as Brahman is everything, everything in manifestation is saguna Brahman.
But, in order to realize that, we pay special attention to the "sattva-saguna-Brahman part", which are sages, jivanmuktas, avataras, devas, srutis, shanti, cit, etc. And we usually call saguna Brahman only to that part (and leave ant, and everything else, out of the term, only to help understanding and realization).
Vivekacudamani, Sri Adhishankaracharya - verse 117
sattvaṁ viśuddhaṁ jalavat tath'āpi tābhyāṁ militvā saraṇāya kalpate yatr'ātma-bimbaḥ prati-bimbitaḥ san prakāśayaty arka iv'ākhilaṁ jaḍam
Sattva is pure. It becomes useful for liberation. Therein is reflected the shadow of the Atman. Sattva manifests the Atman, as the sun manifests the whole of the universe. It is light. Light (sattva), scattering (rajas), and darkness (tamas) are the three qualities
When you think about Brahman as without upadhis only (without maya), without any quality that you can think of, that kind of sunya (void, which actually is not void because is full of Brahman), that you call it nirguna Brahman. So, nirguna Brahman cannot speak, because if He does, then we call it saguna Brahman.
But, as Brahman alone is, He is the only one who speaks, and listen and does everything.
brahmā'rpaṇaṁ brahma havir brahmāgnau brahmaṇā hutam brahmai'va tena gantavyaṁ brahmakarmasamādhinā
For him the act of offering is Brahman, the oblation is Brahman. By Brahman is it offered into the fire of Brahman. Brahman is that which is to be attained by him who realizes Brahman in his works.
Advaitin theory is not that Brahman is without any characteristic, but Brahman is all, and not something in special to exclusion of other parts. There is no two Brahman (Saguna-Nirguna), there is only one, and names are only to understand Brahman, so Brahman is with and without gunas (gunatita Brahman, beyond gunas, with and without)
Vivekacudamani, Sri Adhishankaracharya - verse 243,244
tayor-virodho'ayam upādhi kalpito na vāstavaḥ kaścid upādhireṣaḥ īśasya māyā mahad-ādi-kāraṇaṁ jīvasya kāryaṁ śṛṇu pañcakośam etāv upādhī para-jīvayos tayoḥ samyaṅ nirāse na paro na jīvah rājyaṁ narendrasya bhaṭasya kheṭakaḥ tayor apohe na bhaṭo na rājā
Though they are of different natures (jiva and Brahman), yet in their proper essence, they are equal. What difference you see is only in upadhi, or qualifications (gunas). The upadhi of Brahman is Maya which is the cause of Mahat and its works; and the upadhi of jiva is the five koshas (bodies) and their works. Both are one substance. This substance plus Maya and Mahat is Brahman, and the same substance plus the five koshas is the jiva. Take away the upadhis of both, and what is left is the same thing....