Leaks on the top of black box of culligan water softner under sink system

That's the drain for the reverse osmosis system. It should be tied into a waste line under the sink.
RJH2O

First check the eductor assembly. Top of valve under chrome plate. Pull it out and clean it (after bypassing and depressurizing the system)
If that is OK the the brine piston will need to be replaced. There is a rebuild kit for this series valve that is about $120 that includes the refill piston and a retrofit kit. It would be wise to rebuild the valve.
See your local culligan dealer
RJH2O

If you have an air gap faucet and the water is coming out there. The drain, for the RO, is plugged up or the drain line is pinched.
RJ

This system should be under warranty still. If it is running to drain constantly bypass the system and call Culligan for service. Unplug it for 2 minutes and it may reset.
RJ

Too much pressure in holding tank will not cause leaks in the RO unit. All too much pressure in tank will do is cause low or no product water in RO tank. Older Culligan RO systems (H8, H83, H5) had fittings that used metal locking washers and these can cut the RO line after the RO has been moved around while changing filters. Newer styles (AC30, Water Tower) use quick connect fittings and the RO line may work loose after filter changes and system being moved around under sink.
Turn the incoming water off to RO, turn the water off on RO tank, open the RO faucet to relieve pressure and pull the line off feeding the RO tank, inspect (you may want to cut off a small portion of the line at the filter) check the o-ring and re-install.
RJ

Hello,

They could be loose, try to tighten them a little bit :)

The feed water goes in the bottom and the product water comes out the top to faucet.
Here is a link to Culligan for the owners manual
http://www.culligan.com/en/service-and-maintenance/customer-support/owners-guide/obsolete-manuals/
RJ

It is most likely running to drain. This is an indication that the seal pack is seized up, generally from a bent piston. You need to by-pass the softener and call Culligan for service. If you are handy there is a retro-fit kit available from Culligan to remedy this problem. It takes about 1/2 hr to install.
RJ

replace filter or bfore you go buy filter take out the filter and run the water to see if that salves it and if it does then change your filter

Normally when it leaks out the weep hole the drain line is plugged under the sink....turn off the ro to where it's not making water, disconnect the drain hose and flush it out....make sure the adapter that's connected to the drain pipe is still lined up with the hole if it was moved the water can't get to the drain.