RAW: What we know
Bugbear PC's, as of MotM, are size Medium. And as you note, they have the trait Sneaky, which allows them "without squeezing, [to] move through and stop in a space large enough for a small creature."
As Groody explains in their answer, both Medium and Small creatures fit into the same size space. Thus, as written, any Medium-sized creature can already move through and stop in a space large enough for a small creature, without squeezing, and on the face of it the second part of the of the Sneaky trait actually does nothing for a normally-sized Bugbear.
Suppose, however, your Bugbear has been Enlarged, or has used Giant's Might, and is now size Large. Neither of these changes has altered the Bugbear's race or type, though - and so its Bugbear traits are still active. This is in contrast, for example, with a polymorph spell or a wild shape druid class ability. These might change your size to Large, but they would also change your type to Beast and suppress your Bugbear traits.
Thus, with either Enlarge or Giant's Might in place, you will have a Large Bugbear, who still has Sneaky, and will still be able to "move through and stop in a space large enough for a small creature" without squeezing. Note that your Sneaky ability does not say it works for 'one size smaller'; rather it specifically says it lets you move into spaces permitted to small creatures. Thus, even though you are Large, and would normally not be able to enter a 5' x 5' space without squeezing, you can now do so.
Further, suppose you were already in a 5' x 5' space. Both Enlarge and Giant's Might won't increase you if doing so makes you too big for the space you are in. By some interpretations, that means you could grow to Large in a 5' x 5' space but would be squeezing, but under other interpretations you could not grow past Medium - you would only grow until just before you started to squeeze. The RAW here are not clear. However, regardless of how you as a DM interpret this environmental limit to growth, a Bugbear with Sneaky can grow to Large in a 5' x 5' space; they will not be squeezing, and their growth won't be hindered.
Even more impressive will be if you use Giant's Might first, and are then Enlarged (it won't work in the other order, since Giant's Might will only get you to Large). Under the effect of both of these, you will then be Huge, and still a Bugbear with Sneaky, and thus able to fit in a 5' x 5' space without squeezing. Normally a Huge creature can squeeze into a 10' x 10' space but is not permitted into a 5' x 5' space; you would not only fit in one, but you would not be squeezing, even though you were Huge.
If your Huge Bugbear moved down a hallway of varying widths, you would see an interesting phenomena whereby their sphere of control could contract all the way down to 5' wide, but would immediately expand up to 15' again as soon as space opened up (and their reach would be 5' further on each side of that), with no penalty to their movement as they went.
What kind of sense does this make?
An important Lore shift has happened with Monsters of the Multiverse. Bugbears are now explicitly fey creatures.
Despite their formidable build, bugbears are quiet skulkers, thanks to a fey magic that allows them to hide in spaces seemingly too small for them.
Thus, their ability to fit into small spaces is explicitly magical. A Huge bugbear easily fitting into a small space and suffering no penalties for it is more than passing strange, but should be understood as blurring the boundaries in a way that the fey are known for. It actually reminds me of what first edition did with the Firbolg Giants, who were giantkin that could make themselves small at will.
You might consider cover
Even though Sneaky is explicitly about the 'space' a creature takes up, one of your jobs as DM is to adjudicate Cover. Your descriptions of the size of objects and the size of creatures using them are condensed into your rulings on how much cover a creature has. If a two-foot high barrel gives a four-foot tall goblin half cover, would that same barrel give your eight-foot tall bugbear half cover without it going prone? Perhaps, depending on how far you want to lean into their "fey magic that allows them to hide in spaces seemingly too small for them." Maybe they are really good at crouching without losing movement when they stand. Maybe their form blurs in such a situation and their size gets vague.
Not everything is combat
The "three pillars" of D&D are combat, exploration, and social interaction. The Squeezing rules are explicitly for use in combat; they are found in the combat chapter of the PHB and the size of space for a creature is the size that it controls during combat.
However, not all movement occurs during combat, some of it happens during exploration of the environment. In contrast to the movement rules presented in the Combat chapter of the PHB, those presented in the Chapter on adventuring are more vague:
Even in a dungeon, particularly a large dungeon or a cave network, the DM can summarize movement between encounters: "After killing the guardian at the entrance to the ancient dwarven stronghold, you consult your map, which leads you through miles of echoing corridors to a chasm bridged by a narrow stone arch."
Sometimes, as a DM, you will want to present your players with obstacles or hazards that must be negotiated, and which will depend on their size. In such cases, you might want to rule that there is a difference in the ability or ease of passage of small and medium creatures, despite them controlling equal areas in combat. Given the wording of the Bugbear ability, I would strongly suggest that you treat them as a small creatures for such interactions between environment and movement, and do so even if other magic had made them temporarily Large or even Huge.