It is undefined, but I would say Yes
What kind of "can" is it?
As you say, the rules on Breaking Up Your Move state:
You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet.
But also as you say, the Phoenix's legendary action does not take place on its turn. So how should we read this?
We usually read "can" in a prescriptivist way, as a "can only" - 'you can only break up you move when it is your turn'. This is how can is used in "Hit Points and Hit Dice":
After you rest, you can spend Hit Dice to regain hit points
Here, we understand this to mean that spending Hit Dice to regain hit points can only be done after a rest (unless another feature explicitly grants this ability, such as the wither and bloom spell).
But, we sometimes use can in a descriptivist way - 'whenever you have both a move and an action, like on your turn, you can break up the move'. This is how can was used1 in the Choosing a Race section:
Your choice of race affects many different aspects of your character. It establishes fundamental qualities that exist throughout your character’s adventuring career. When making this decision, keep in mind the kind of character you want to play. For example, a halfling could be a good choice for a sneaky rogue, a dwarf makes a
tough warrior, and an elf can be a master of arcane magic.
Here, there is no "only" implied. We do not read this passage to mean that elves can only be masters of arcane magic, nor that only elves can be masters of arcane magic. Here can is just providing more detail on one possibility.
So how should we read can in the case of breaking up your move? By itself, the rule doesn't say either way. And most of the time, it doesn't matter, because it is very rare that a creature would get both a move and an action when it is not their turn (Readied actions, for example, can give you a move or an action but not both, and generally forced movement gives you only a move). But when it does matter, when you have both, such as in the case of this legendary action, what should we do? I don't think there is a RAW answer here, so how your DM decides this will depend on their approach to the game.
What I would do
In deciding this, my basic approach is to give a creature a lot of agency on its own turn, and much less on the turn of another creature. This is, after all, one reason we have turns to begin with, to decide the precedence of competing actions. On its own turn, a creature can perform a number of "flourishes" and embellishments to its movement and actions. I wouldn't permit these when it is not its turn. You can mount or dismount "once during your turn" - I wouldn't permit that special kind of movement to a creature when it is not their turn.
But what creatures can definitely do, even when it is not their turn, are the things that other features explicitly permit them to do. In general, you cannot move when it is not your turn. If something permits you to move when it is not your turn, that thing itself must then spell out the conditions under which you may move. In general, you cannot take actions when it is not your turn, except when some feature permits you to. In this case, what is permitting the Phoenix to move and act is its legendary action. Thus, it is permitted to do whatever its legendary action description says (although only what it says).
The phoenix moves up to its speed and makes one Fiery Talons attack.
The Phoenix is permitted to move up to its speed. Is breaking up its move (to take an action also explicitly permitted to it) somehow more than movement, more than what it is permitted? Has it done something other than just move and attack? I would say not - the pause in its movement to attack is not an 'extra thing' that it did. Does it take more time? I would say not - if it switches immediately from moving to attacking and back to moving, but doesn't move any farther than it could anyway, then it hasn't overstayed its welcome on someone else's turn. It is only moving and taking its action, and the legendary action permits it to structure the order of those as it chooses.
1 At least this is what my print copy of the PHB says. The same section in D&D Beyond, at the time of writing, says:
The race you choose contributes to your character’s identity in an important way, by establishing a general appearance and the natural talents gained from culture and ancestry. Your character’s race grants particular racial traits, such as special senses, proficiency with certain weapons or tools, proficiency in one or more skills, or the ability to use minor spells. These traits sometimes dovetail with the capabilities of certain classes (see step 2). For example, the racial traits of lightfoot halflings make them exceptional rogues, and high elves tend to be powerful wizards.
