When the government of the United States of America reopens, will furloughed government workers be paid for the missed time?
By default, no. The definition of a furlough is that your worker is not working and not being paid for it. If they were being paid, it would be a vacation instead. This is why the one-day-a-week furloughing of some workers is a cost-saving measure - it's just gone up to a one-day-a-day furlough.
That being said, there is historical precedent for Congress to retroactively pay for the missed day. From @user1873's article:
Congress has granted back-pay to furloughed federal workers after each of the past 16 shutdowns that have taken place since 1976.
This is something that has to be done on a case-by-case basis, though. There's a wide range of possible results, since a bill has to be passed to specifically address this shutdown's back pay. It could range from doing nothing (which would result in no back pay), to full back pay (as with the others), or some compromise might be worked out (such as 50% back pay, or the option to have retroactively used vacation days to receive full pay).
They likely will be paid retroactively whether they worked or not.
The House has passed H.R.3235 To provide for the compensation of any Federal, State, or local employee furloughed due to a lapse in appropriations which began on or about October 1, 2013.
(a) IN GENERAL.—Any Federal employee or State or local employee furloughed due to a lapse in appropriations (from the Treasury of the United States) which began on or about October 1, 2013, shall be compensated at such employee’s standard rate of compensation, for the period of such lapse in appropriations, as soon as practicable after such lapse in appropriations ends.
During the mean time, furloughed employees are likely eligible for unemployment benefits. If the Senate passes and President signs the House bill, those federal employees will be paid for the time they didn't work. This was the previous case in the last 16 shutdowns. They are then supposed to repay any unemployment benefits they received. Considering the governments past record of improper payments, about 12% of federal employees will not only receive a 2+ week vacation, but an additional source of income.
Program: Unemployment Insurance, Agency:Department of Labor, Reported Improper Payments: $13.7 billion / 12%, Reported Primary Cause(s):Overpayment to claimants who continue to claim benefits after they return to work, ineligibility, and claimants who failed to meet active work search requirements