Both might be considered emergencies but I think that one is probably more appropriate. I would talk with the employees as a group and tell them that some employees are abusing the policy and there might be changes if things continue. Then I would sit down with the abusers one on one and explain how they are abusing the policy.
Any less than 2 weeks, including sick days, I think I would allow enough unpaid time to equal two weeks. Things happen and people do need time away from the desk. As others have stated, if people are abusing it, they dont' respect their career and fellow co-workers.
I wouldn't limit it yet if employees haven't been warned. Give the warning that continued abuse will lead to limits for individuals and possibly termination and let employees hang themselves.
Obviously this can bring down the company and coworkers but first off you need to talk to those that are abusing it and figure out if they are burned out, have something going on outside of work that is taking up a lot of time (family issues), or if they are just slacking. IMO you can't make any true judgement on the issue until that point and it has to be dealt with on a case by case basis. If they are abusing it for no real reason or are dragging the group down and there is no way to figure something out than you have the right to let them go.
Just remember they are people not just workers.
If they want unpaid, really make it unpaid by having them pay for their benefit package for that time off. Then you can waive that penalty for legitamate cases.
Have they exhausted all their vacation days yet?
I have a client that offers leave without pay, but that number acrues throughout the year, and every time they reach another factor of 10 (I think, maybe more) in unpaid leave, they don't get vacation time for that pay period and (I think this next part is accurate, at least it used to be but it may be illegal now) and have to pay for all their benefits and no 401k match. It may not be all of that, but its some combination of all that.
That, and preapproval of all unpaid vacation. If you allow your employees to accumulate vacation days, then short of a long medical issue or having a baby, there's no reason they should have to use unpaid leave.
Is this a job that doesn't offer paid time off? I've worked plenty of jobs that simply didn't have paid time off. If you needed time off, you just ran it by the boss and they either said yea or nay. Granted these weren't "career" type jobs by any means, but I haven't seen anything in the OP to suggest that these are the type of employees that already have PTO.
Not needed unless its their first year. Then they have to get approval and then they just don't get paid.
This included legitimate cases (maternity, sick leave) since those would be covered by short term disability which typically doesn't kick in until day 5 or 7 to give the employee some $$, so the employee would be unpaid until their short term disability kicked in.