In response to the additional information, if there are no lapses in wage, then there should be no problem in extending your H1B UNLESS the employer is H1B dependent/willful violator/TARP recipient AND that you are not an "exempt" H1B nonimmigrant (meaning you have a masters or at least a $60K salary) AND the employer displaced (laid off) an essentially equivalent US worker (one whose job responsibilities are essentially the same as yours) within 90 days of filing your LCA (which is required in order to file an H1B petition). These rules are found in 20 CFR 655.736-738.
So assuming your employer is H1B dependent because it has hired more than a certain threshhold of H1B workers (or it was a past willful violator or TARP recipient), and that you are not being paid $60K or have at least a masters degree, and of the 30 layoffs, one of those employees had essentially the same job as you, then the employer would not be able to file an LCA (and the H1B extension) for you until 90 days from the date of the layoff.
I am not aware of a rule relating to furloughs, however.
Finally, you indicate there are only 3 H1B workers, so the company is clearly not H1B depedent. So unless the company was ruled as a prior willful violator or TARP receipient, you should be able to go forward with the extension now.
I think your employer is just telling you it is legally unable to apply for visas or extensions during furloughs for the sake of damage control.
There are anti-benching provisions under H-1B rules, and this includes furloughs. This means that the employer must pay H-1B workers in full, even during nonproductive periods UNLESS the worker is unable to work due to conditions unrelated to employment (20 CFR §655.73(c)(7)(i)). If an H-1B employee’s salary will fall below the required wage, then the reduction will not be in compliance with DOL regulations. In this case, the choice becomes either not to pursue the furlough, or to terminate the H-1B employee in order to remain in compliance.
So basically, your employer violated DOL regulations for putting H-1B workers on furlough instead of firing them (in cases like this, while I sympathize with the employee, sometimes the employer is just being nice to not terminate the worker, and it looks like your employer is going through a rough patch). Penalties can include paying back unpaid wages to workers (H1B workers can pursue claims with the DOL), fines from $1,000-$35,000 per violation, and to be deemed a willful H-1B violator can make it much more difficult to apply for H-1Bs in the future (it may subject them to pre-filing recruitment requirements).
I hope this is helpful.
I should add that, in this situation, the employer can file an extension.
USCIS will expect to see the most recent pay-stubs as proof of your maintenance of H1B employment. They expect that you have received prevailing wages for the specific occupation. If there has been a furlough where wages weren't paid, the employer should show that it compensated you for those unpaid wages before filing the H1B extension.
The rules I listed in my previous answer are probably what the employer is afraid of. They might fear that filing an H1B extension where their furlough of H1B workers is exposed may trigger a DOL audit. I've never personally seen an audit triggered because USCIS finds a lapse in an employee's wages in an extension request. USCIS may forgive minor wage lapses or it might not. If the lapse is not forgiven, USCIS may still approve the petition, but it could require you to go for consular processing to get back on H1B status.
If the company says the are unable to extend your H1B, and that you will have to leave, they are obligated to pay your airfare.