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Balrogs adopt some physical form, which they can change at will

From Tolkien's essay Ósanwë-kenta, included in Vinyar Tengwar #39:

Melkor alone of the Great became at last bound to a bodily form; but that was because of the use that he made of this in his purpose to become Lord of the Incarnate, and of the great evils that he did in the visible body. Also he had dissipated his native powers in the control of his agents and servants, so that he became in the end, in himself and without their support, a weakened thing, consumed by hate and unable to restore himself from the state into which he had fallen. Even his visible form he could no longer master, so that its hideousness could not any longer be masked, and it showed forth the evil of his mind. So it was also with even some of his greatest servants, as in these later days we see: they became wedded to the forms of their evil deeds, and if these bodies were taken from them or destroyed, they were nullified, until they had rebuilt a semblance of their former habitations, with which they could continue the evil courses in which they had become fixed".

Never in the legendarium do we see a balrog change its form, and this is probably why - they weren't able to, and like their master, were trapped in their form of power and malice.

We also know that in early drafts of the legendarium, "Melko" specifically kidnapped eagles to experiment on because he was unable to replicate flight. That's part of why the flying wyrms were so surprising and devastating when he finally unleashed them, but balrogs were created long before, and we can easily conclude that, therefore, they were not created with wings.

(Also, if Gandalf could fly, he wouldn't have needed Gwaihir to rescue him from the pinnacle of Orthanc. Tolkien's legendarium isn't Dragon Ball Z.)