Well with either of these moves white can just play e4 and it becomes a king's pawn opening.
This is now the owen defence, a dubious but playable opening. If you play 2. ...d5 then it's the French. Against c4, e6 b6 is the English defence. It's also a dubious but playable opening. It's nowhere near as reputable as the Queen's Indian which is e6-b6 after Nf6 which prevents e4.
English defence:
If e4 isn't played on move 3 then f5 is the idea and the only real reason to delay Nf6. The disadvantage is that it simply allows e4.
This is the Queen's Indian:
This is the sister opening to the Nimzo-Indian which is played against Nc3 instead of Nf3. This is an incredibly solid and respectable defence. One of the best, in fact.
I'm assuming since you talk about starting with e6-b6 that maybe you want something offbeat but I would much more recommend the Nimzo/QID.
anyone know where to look for info about 1e6 2b6 as black or the modern with d4? also open to alternative openings against d4.