A typical road frame could have these issues for cyclocross (CX):
- tire clearance -- cyclocross tires are a bit bigger than typical road tires (especially with the small knobs typical on CX tires)
- brake style -- you'll want some kind of brakes that aren't clogged up by mud. Road bikes often have caliper brakes that hug the wheel fairly closely, while cyclocross bikes usually have cantilever (or sometimes V-brakes). Disc brakes are recently allowed in CX and seem popular on the higher-end bikes.
- frame style -- a lot of newer road frames have a compact geometry where the top tube slopes down to the seat cluster. Cyclocross bikes generally have a straight top tube so that it's easier to pick up the bike and carry it with your shoulder under the front of the seat cluster.
- handlebar height -- not as important, but cyclocross bikes generally have slightly higher handlebars since aerodynamics aren't very important in cyclocross
- gearing -- typically run a little lower. singlespeed is popular (but probably puts you in a different class than geared racers)
There's some even more minor things, like some CX bikes have a slightly higher bottom bracket than road bikes to make it easier to get over rough terrain and cables are probably routed away from the muddy side of the bike.
I wouldn't go with the smaller frame, as that'll likely be less comfortable, effectively lower handlebars, and probably less room for your shoulder if you need to carry the bike.
If you don't already have a bike you'd be better off getting a cyclocross bike with two sets of tires (one knobby and one smooth) and a second set of gears and converting it to a road bike part-time, instead of getting a road bike and using it for CX.
(I haven't raced cyclocross, but have watched a few races, including one a couple days ago and am starting to consider trying it out)