Can't answer the overall question, but for Group 1 specifically, there's a discussion of the recruitment program here. The original plan was to recruit twelve, to allow for some dropping out during the program. When it became clear there were unlikely to be many dropouts, they revised the target to six. However, when making the final selection, they got stuck on bringing the shortlist down to 6:
Sitting in judgment over 18 finalists, Donlan, White, and North pared down the final pool of selectees, choosing each to complement the rest of the group. The going was so difficult that they could not reach the magic number six, so Gilruth decided to recommend seven. Donlan then telephoned each of the seven individually to ask whether he was still willing to accept a position as a Mercury astronaut. Each one gladly volunteered again.
So that one's as much an accident as anything...
(edit:) Group 4 was originally six strong, but one (Graveline) resigned before being assigned to a mission.
(edit:) Group 7 was essentially arbitrary - it was "all the Manned Orbiting Laboratory astronauts under 35". It so happened there were an odd number of them, but there was no set number initially aimed for.
After accounting for these, we end up with:
- Planned even, actually odd (7)
- Odd (9)
- Even (14)
- Even (6)
- Odd (19)
- Odd (11)
- No plan for numbers, actually odd (7)
- Odd (35)
- Odd (19)
- Odd (17)
- Odd (13)
- Odd (15)
- Odd (23)
- Even (24)
- Odd (23)
- Even (44)
- Even (32)
- Odd (17)
- Odd (11)
- Even (14)
- Even (8)
So across all 21 groups, and taking account of the fact we know two were intended to be different sizes, this is eight even, twelve odd, one "whoever happens to qualify" (but happened to be odd). This certainly feels like there's no intentional focus on odd numbers.