Is my employer responsible for my ambulance transport bill if I got sick at work?

Why would your employer be liable? To analogize: if someone fell ill at your home, for example, would you automtaically be responsible for their amubulence bill and other medical costs? The answer is no--without more, neither you nor your employer would be liable.

A person or business is NOT responsible for another's injuries or medical expenses unless they were at fault in causing them; the mere fact that someone because ill or was injured at a place of business (or at a private home) does not make the person or business owing or renting the property liable.

The fact that you were "pressured" does not make the employee liable--there is no fault in "pressuring" an employee to reach out to previous client. Pressure is a natural part of work; if a given job, career, or industry is too high pressure for a particular person, that person probably needs to consider a career change; however, the employer is not responsible for any stress-induced ailments or their costs.