How do you evolve Budsquire?

A Budsquire that is level 27 but has the wrong personality, or has company, will not evolve.

All three conditions have to be true together. Take every other Aniimo out of your party so Budsquire is on its own, and check that its personality is Faithful. A Budsquire without Faithful does not meet the requirement at any level, so check the personality first. Otherwise you can level one to 27 and get nothing.

Thornblade or Melobloom

Budsquire has two evolutions. One is Thornblade and the other is Melobloom.

The Melobloom route also has a level 27 requirement, and on top of that it costs one rose stamen, the Stamen of Kubo's Rose. Have that item ready if Melobloom is the form you want. Melobloom is a Grass support with unusually high Break for a buffer, and the linked article covers its full kit.

Not settled: The game does not make clear which of the two forms the Faithful, solo party condition produces. It also leaves open whether the Stamen of Kubo's Rose route to Melobloom needs Faithful and a solo party as well.