Paying premiums for new employees

Small employer group health policies have, since ACA, been allowed to impose no more than a 90-day waiting period on new employees before they must be offered coverage.  42 U.S.C. 300gg-7.

Can a small employer offer coverage in that time frame to a new employee, but it would require the employee to pay all of the premium expense until say 1 or 2 years of employment have passed?

In theory that would be allowed.  But keep in mind that a) the carrier probably has minimum employer contribution requirements, b) if this is an ALE, they would be exposing the company to "B Penalty" liability because the offer would be unaffordable for most, and c) the Section 125 uniform election rule may present an issue from an NDT perspective (overview here: https://www.newfront.com/blog/designing-health-plans-with-different-strategies).

Also note that you can have a substantive eligibility condition of up to one month (referred to as a "bona fide orientation period") before application of the 90-day waiting period.  You just have to be careful to coordinate that with the employer mandate limited non-assessment period rules if it's an ALE. More discussion here: https://www.newfront.com/blog/aca-first-day-of-the-fourth-full-calendar-month-rule