It all depends on the Terms and Conditions you signed when you joined the company.
To be fair it does sound a bit ridiculous that you have to get a visa on your own time to be able to do a job the company employed you to do.
A few moons ago I worked for a charter company based in the uk. Working conditions tend(please note i said TEND, to those who might misconstrue) to be alot worse working in charter than it is working for a decent scheduled airline.
However when it came to the american visas the company gave us(everyone who went on to long haul) the appropriate documentation to fill out, the company made the appointment for us at the embassy, the company paid to coach us to the embassy and back, and the company paid for the visas. This was all done on a rostered working day and we were paid our hourly rates/allowances for the whole day on top of that.
It sounds to me like your company is seriously taking the p***.
If I were you I would sit and read the company T & q's again. Maybe question someone in hr or take it up with the union.