Here's the description of the boots of teleportation:
Any character wearing this footwear may teleport three times per day, exactly as if he had cast the spell of the same name.
Since it says the effect is exactly like the teleport spell, we can just look at that spell's description:
This spell instantly transports you to a designated destination, which may be as distant as 100 miles per caster level. Interplanar travel is not possible.
Well, that seems entirely clear and biguous: No interplanar travel is possible when using boots of teleportation. In fact, even if that line wasn't there, you'd only be able to teleport between planes in highly specific circumstances: Another plane can only be within 100 miles per caster level if both the origin and destination planes are coterminous or coexistent, and most arbitrary pairs of planes aren't.
Incidentally, your question briefly mentions "greater" boots of teleportation. I think this might be an error on your part, as Google can't find any reference to such an item. If it's not an error, this item must be either homebrew or found in a supplement I'm not familiar with. Either way, I strongly suspect that it allows you to teleport as if you'd cast greater teleport instead of teleport - in which case, your player is still out of luck, as that spell's description explicitly states:
Interplanar travel is not possible.
Yup, greater boots don't allow interplanar travel either. Your player will have to charter a ferry up the Styx like the rest of us.
As for whether the boots work on planes other than the Prime, teleportation spells work anywhere with a connection to the Astral plane, because teleportation spells work by instantaneously transporting the transported individuals into the astral and then back out in a different location; As long as the Astral's connected to both points, and the spell doesn't put any other restrictions on where you can go, teleportation can work. (There's a whole pile of complicated lore about this scattered across multiple editions, but it basically boils down to "the Astral has neither distance nor time, but it has the concepts of both, and this can be exploited if you know how.")
Access to the Astral isn't a problem on most Prime worlds, but things get weird in the multiverse beyond. Specifically, there's an obscure line in the DMG (but not the SRD) that specifies only the first layer of each outer plane has a connection to the Astral. This means that if you want to teleport within the Abyss, you'd better hope that both your origin and destination are on that plane's first layer; Otherwise, you'll need to walk. Or possibly wade. Best not to ask through what.