AFAIK, you have to finish BMQ, BMQ-L (SQ), and DP1 and have 1-2 years of on job experience before you can advance to any specialty training course in any trade.
However, to correct you on the "a lot of people tend to shy away from jobs like EOD" is totally the opposite of truth, EOD is one of the most desired specialty training in Combat Engineering, pretty much like Sniper course for Infantry. at least in my unit thats what i seem to gather, almost everyone would like to catch a chance at it.
p.s. i'm in Reserve, not Reg