Can my employer require that I use a fingerprinting machine?

It is legal for your company to require a fingerprint machine in order to protect its intellectual property. Further, it is generally legal for your employer to put a new company policy in place and fire you if you don't comply. 

At will employees can be dismissed for any reason, which means it is perfectly permissible for your employer to dismiss you from your job should you refuse to participate in something like a fingerprinting machine used for identification purposes. Whatever your reasons might be for your refusal to follow company policy, your employer does have the right to expect you to meet expectations and participate in the usage of the new security system. The use of the new system may be an integral part of operations at the company. What's more, in the majority of states, because of “at will” employment laws, your boss can fire you with very little explanation or justification at all, provided her or she is not breaking any laws by acting in a discriminatory fashion against you. The exception to this rule is when the reason that your employer has for firing you is illegal. For example, it is against federal law to discriminate against an employee on the bases of race, religion, creed, or national origin, and therefore if your employer fires you based on any of these things it doesn't matter whether you are an "at-will" employee or not, the action is illegal.

Refusing to use a company-wide security system being implemented onto the office computers may make sense to you, but the company is making this change for a reason within their business plan. As an employee, you certainly have a right to choose not to participate in it, but your boss, in turn, has the right to consider any employee who does not accept the new system as uncooperative and worthy of dismissal.