I think it's time we put the phrase "I have an unspoken prayer request" out to pasture. Seriously, let's just kill it.
It's one of those ideas that keeps us hidden from the people we're supposed to be visible to. When you say that to someone, you're hiding. You're putting a wall between you and the idea of prayer, which I think is meant to be vulnerable. You're hedging your bets in a way, asking for someone to give you something without giving them anything at all. It's a little hollow and a little empty and a little fake.
You can always go to James 5:16 to get a reminder of what God says about confession:
Therefore confess your sins to each other and pray for each other so that you may be healed. The prayer of a righteous man is powerful and effective.
It doesn't say, "sort of confess," it just says confess. So I think that's what we're supposed to do when it comes to asking someone to pray for you.
And here's the other thing, when you tell someone you have an "unspoken prayer request" people assume the worst. They might not admit it, but people automatically think, "wow, that dude is having an affair" or "yikes that guy is embezzling money" or something equally as big.
So do the simple thing. Ask for prayer with words that mean something and lets retire the phrase, "unspoken prayer request."
p.s. Be careful about James 5:16, it's one of those verses people often use as an excuse to employ honesty as a weapon.