A very important thing to note here is that when an entangled pair snaps to one state or another, both parts of the pair do it simultaneously; however this can't be used to impart any information unless you have already passed on the information in a classical manner.
What this means for your race of distance breeders is that they'll either have already had some form of physical contact, or they're going to have to transmit the information via EM waves. At that point, you may as well forget the entangled pairs, because it's going to be considerably more effort for a race to set up and maintain pair entanglement (I'm not even sure how you'd do this without having some form of physical contact) that is then turned into usable genetic information by some classical interaction than it is to just rely on the classical interaction in the first place.
One possibility that maintains breeding at a distance but doesn't rely on entanglement is that the 'egg' of your species is sensitive to certain frequencies of the electromagnetic spectrum. The egg created by the mother contains half of the 'genetic' code (lets call it the Base) and a series of unpolarised (seems like a good enough word) chemicals that can then be pushed into various forms by the correct application of... ooh, lets go for Gamma radiation, because it sounds cool. Use neutrino if you want it to be nigh on undetectable, or visible light if you want it to look nice.
When a female goes into heat, they settle down and send out a pulse containing their Base information, encoded in a radiation pulse. A nearby male receives this pulse, and sends a pulse back encoding all of their genetic information, keyed to the mother's Base information. This then polarises (fertilises) the egg with that mother's specific Base, and forms babby. The two way mother/father communication will cut down on environmental fertilisation (which could be catastrophic).
TL:DR: There is no reason to use quantum entanglement, because you'll need something classical to transmit the information anyway.