No. The above idea naturally is unevidenced & unsubstantiated . If the state of enlightenment was free from concepts then when the Buddha was teaching others Dhamma using concepts his mind was not enlightened. In Theravada, the state of enlightenment is free from craving, atttachment & self-view, as clearly explained in the Pali suttas.
The Buddha-To-Be already practised non-conceptual meditation with Uddaka Ramaputta & Alara Kalama and intuitively understood these were not Nibbana (due to their temporary nature).
Only the end of craving & attachment is permanent enlightenment. Non-conceptualisation cannot be permanent enlightenment. This is impossible. The Buddha described enlightened as follows:
Dispassion is said to be the best of all things whether conditioned or unconditioned. That is, the quelling of vanity, the removing of thirst, the abolishing of clinging, the breaking of the round, the ending of craving, fading away, cessation, extinguishment.
Yāvatā, bhikkhave, dhammā saṅkhatā vā asaṅkhatā vā, virāgo tesaṃ aggamakkhāyati, yadidaṃ madanimmadano pipāsavinayo ālayasamugghāto vaṭṭupacchedo taṇhākkhayo virāgo nirodho nibbānaṃ.
No. SN 12.10 describes the Bodhisatta making a breakthrough to enlightenment using concepts, as follows:
On a more subtle level, enlightenment is actually conceptual anyway. For example, prior to the enlightenment, there was no experience of not-self or anatta. But during & after enlightenment, the mind develops a knowing & knowledge of anatta. Then after enlightenment, the mind is able to explain not-self to others. This shows a very subtle level of "conceptualisation" or "describing" is occurring with enlightenment. "Not-self/anatta" is discerned & inwardly described by sankhara khandha (which includes the faculty of thought or conceptualising).Then, monks, this thought occurred to me 'What being present does decay-and-death come to be? What conditions decay-and-death?' Then, monks, as I considered this thoroughly, the insight and comprehension dawned on me: 'Birth being present, death-and-decay comes to be; decay-and-death is conditioned by birth.' Then the thought occurred to me: 'What being present does birth come to be? What conditions birth?... becoming... grasping... craving... feeling... contact... the six sense-bases... name-form... consciousness... formations? ...' Then, as I considered this thoroughly, the insight and comprehension dawned on me: 'Ignorance being present the formations come to be; the formations are conditioned by ignorance.' And so we have it like this: 'Conditioned by ignorance are the formations, conditioned by the formations is consciousness... So there comes about the arising of this entire mass of suffering.'
"'Arising, arising!' — At this thought, monks, there arose in me, concerning things unheard of before, vision, knowledge, understanding, light.
There is no such thing as "mindfulness of breathing". The word "sati" means "to remember". Present moment breathing is not something remembered. What is "remembered" is something from the past, as follows:
And what is the faculty of mindfulness? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous, remembering & able to call to mind even things that were done & said long ago.
Dwelling thus withdrawn, one recollects that Dhamma and thinks it over. Whenever, bhikkhus, a bhikkhu dwelling thus withdrawn recollects that Dhamma and thinks it over, on that occasion the enlightenment factor of mindfulness is aroused by the bhikkhu; on that occasion the bhikkhu develops the enlightenment factor of mindfulness; on that occasion the enlightenment factor of mindfulness comes to fulfilment by development in the bhikkhu.
One is mindful to abandon wrong view & to enter & remain in right view: This is one's right mindfulness