Currently only "essential" government employees are allowed to go to work and they're doing so without pay. But as the shutdown gets longer and people's savings start running out, would anything prevent these employees from quitting their job? If not, could this mean that eventually even essential government services will be forced to shutdown due to lack of personnel?
In most cases, no
Most government employees are covered by the same "at-will" employment rules as any other employee at any private firm in the United States.
In brief, at-will employment means that either you or your employer (subject only to certain restrictions like anti-discrimination laws for example) are free to terminate the employment relationship at any time, for any reason, with or without notice.
Some accrued benefits like back pay, unused vacation time, and vested retirement benefits must be paid to the employee by law, but other benefits, such as pensions and bonuses may be forfeited according to agency policy.
Although you normally don't qualify for unemployment if you quit your job, many employees who quit under such conditions may still be eligible for unemployment benefits. Unemployment law generally allows you to receive benefits if you quit "due to an unconscionable act" by your employer (like not getting paid for hours you legitimately worked), but this normally involves a hearing in which you must plead your case. Even if you win, you'd still be required to pay those benefits back if/when you eventually get paid for the hours you received the unemployment benefits for.
The only exceptions to these rules are military personnel and employees covered by an employment contract. For military personnel, failure to report to one's duty station without authorization is a criminal offense (they are AWOL at that point). As for contractors, they may be subject to civil liability depending on the terms of their contract. In both those cases, however, severance restrictions are spelled out in the contract/enlistment documents. So those people are still considered voluntarily employed, even if they aren't free to quit.