If the employer has instructed you to lift heavy boxes without having given you any manual handling training, then they are in breach of the Manual Handling Regulations, and you will have an entitlement to claim damages from them.
Ethel is right that your full-time employers may seek recompense from you for any sick pay that they have paid you while you've been off sick, but this can be included as part of your claim against the part-time employers. Although you say that the injury isn't permanent and should heal within a few weeks, the fact that you have had to spend a day in hospital and take time off work means that it will more than likely be severe enough to beat the Small Claims limit on claims like this (currently �1,000).
If I were you, I'd speak to a Solicitor that offers No Win No Fee.