What happens to accrued sick leave once a company is sold?

It depends how the company was sold. If the company were an LLC or corporation (which it most likely was if it was a "large healthcare company") and someone else bought that LLC or corporation, then you should still have the same sick leave, since you are working for the same entity (same LLC or corporation), just with a different owner of it.
If the new owner did not buy the LLC or corporation--for example, they bought the assets and right to continue the business and hired the employees, but did not buy the existing corporation entity--then the new owner does not owe you any of this leave time, since in this case, you are now working for a new entity (like working for a different person).
If you were laid off or terminated as part of the sale, you lost any unused sick leave: there is no right to be paid out for unused sick leave on termation of employment. (With one exception: if you had a written contract or union agreement requiring them to pay it out, they have to follow the contract and do so--if they do not, you could sue them for the money.) 
If you were terminated or laid off, they would have to pay you unused vacaton time *if* there was an employer policy to pay these things out on termation, etc.--such as policy could be found in an employee handbook or policy manual, or even in the employer's past practice (have they paid other employees?). But if their policy/practice is clearly to NOT pay vacation on termiantion, they do not have to do so.