Laces do stretch in the beginning. But I've never needed a break-in period in this sense: If I use new laces on a well broken-in pair of boots, I've never had to re-lace during a session because the laces had stretched substantially during the session. The next session, the laces are stretched from their original length, but that's automatically taken care of during the new lace-up. How much stretching occurs depends on the laces: e.g., the Riedell cotton/poly laces don't stretch that much; the Jackson poly laces stretch more. But they're usually stable after 3 or 4 sessions.
If your question is whether you should intentionally use broken-in laces to break in a new pair of boots, the answer is no. Makes no difference. You will need to re-lace several times during the first several sessions anyways, but that's because the boots are new and need breaking in.
If your question is whether it's OK to use broken-in laces to break in a new pair of boots, the answer is yes. That is, if the laces are of the right length, in otherwise good condition, and you would have no other use for them anyway, no need to chuck them, just go ahead and use them.