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All counters, even -1/-1 counters stay on a creature until a spell or ability removes it. +1/+1 counters DO NOT remove -1/-1 counters from a creature. Fate Transfer would be an example of a card that can remove counters. Quillspike is much better though because it only costs 1 G/B to remove a -1/-1 counter and it gets +3/+3 until end of turn.

"+1/+1 counters DO NOT remove -1/-1 counters from a creature."

this is incorrect. -1/-1 counters and +1/+1 counters DO nullify each other.

if your creature has 3 +1/+1 counters on it and it takes damage from a wither creature with a power of 2, your creature would end up with one +1/+1 counter remaining on it.

source: one of the 'rule tips' cards included in booster packs. I believe they also cover this in one of the planeswalker videos.

caveat: remember that +1/+1 counters and -1/-1 counters are not the same thing as "creature gains +1/+1 till end of turn" and "creature gets -1/-1 until end of turn". These types of effects do NOT interact with the counters.

so, to answer your question: YES, you can use anything that adds a +1/+1 counter(or more) to a creature to 'reboot' it's persist ability by removing the -1/-1 counter on it.

From the rules:

"120.2. If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it as a state-based action, where N is the smaller of the number of +1/+1 and -1/-1 counters on it."