My tax software suggested claiming 11 for more taxes to be withheld vs claiming 0? How would this work?

The answer depends on many personal/family factors and preferences. I am sure you're receiving a big refund, that's why the pre-programmed tax software's sending you a typical automatic tax advice message. You need really check out a W-4 form (print it out from irs.gov), if you think that it might be necessary to change your withholding allowances. Actually, fewer the number you'd claim on W-4, more money is taking out (withheld) from your paychecks for federal/state taxes (means paying taxes in advance) and that's also increasing your chance of getting back those money you've paid in advance, when you file your taxes if you no tax liability and/or if you qualify some tax credits to offset your tax liability. "You can take your projected tax credits and deductions, exemptions into account in figuring your allowable number of withholding allowances. If you have little kids, you may have child or dependent care expenses and the child tax credit. Earned Income Credit is other very important credit, although there is certain income limit to qualify"