I would strongly advise against it. I've been through some competitive application processes, security clearances, polygraph exams and it's all very unpleasant. I've been a part of the selection process and cannot believe the petty things qualified applicants are dropped for. There is usually a formula and you may be the highest ranked applicant in the pool. Then some socially maladjusted grad student without a friend in the world says they think the tie in your picture is annoying... bam, do not rank list.
Most applications even for even basic jobs now are prescreened by third-party consulting firms that dig up every shred of data inexistence on the person. It's amazing what they can track down. I wouldn't make a breadcrumb trail to anything that could be considered conduct unbecoming. Once you're in people don't pay much attention anymore, but the last thing you want is a red flag in a place that every employer will scrutinize like Facebook. A lot of furs are fairly highly educated and in the early stages of that education. You never know what might come of the future. Med schools/ academic programs don't dig much deeper than a criminal background check. However when you’re done and applying for that job, whether Skunk Works or Wal-Mart, you may have an issue.
I have some friends that had success doing what Gibby is suggesting. They have a Facebook page linked to their name that is squeaky clean and an alternative identity on Facebook that is their personal life but it's not linked to their identity.
There are law-enforcement, fire, military furs who make no attempt to conceal their identity on FA. I would be curious to know if it's ever come up in the hiring process for them as screenings for those positions are generally fairly probing and wildly inappropriate.
And then there's the entirely separate matter of busy bodies that go goggling you... Heck I've even done it before interviewing people. It's just so darn easy.