Bonus pay period start and end vs check date

If a bonus is paid out to a participant after they have already entered the plan, but the period for which the bonus compensation was earned was before they had entered the plan is this eligible compensation to defer? 

Yes, you can defer on compensation that is not yet available to you (i.e., pay date), but read and follow the plan document as Bri alluded.

I agree with Bri and CuseFan.  Even though the bonus was attributable to the employee's performance for a period predating his/her entry into the plan, for the most part, in the absence of a contrary plan provision, what is considered eligible compensation is determined on the cash receipts and disbursements method (i.e., salary and bonus amounts are considered to be compensation when they are paid even if  earned prior to that time. Most payroll systems tend to follow that method. 

You need to check the definition of compensation in the plan document.  I can see two options.  First, the bonus amount is included for the plan year in which it is paid.  In other words, the bonus for 2022 is actually paid in March 2023 and will be taxed in 2023.  This would be considered 2023 compensation.  On the other hand, the plan could be written so that the 2022 bonus is included in the 2022 compensation.  This would seem to muddy the waters, but it would be permissible.    I would expect this for some high-level employees, such as members of the board of directors, who are covered by a non-qualified plan / executive plan.