Back to your problem. I have read that these small mills really need a LOT of lubrication if used regularly and regular CNCing would be classified as a lot of use. The machine should actually be sitting in a pool of oil tramp oil. This oil should be dripping from the machine. Basically if it's not, the machine is under-lubricated.
The spindle drive is via a number of plastic gears and these need to be kept well greased, any good quality light grease should work fine. I've greased mine. But with CNCing, you need to schedule-in regular lubrication [depending upon your usage of the machine] - say weekly or fortnightly. The oilers need to be checked daily and topped-up before use.
It's worth adding oilers and grease nipples As necessary. I was considering A couple of oilers on a bracket [mounted on the head] connected via flexible small bore plastic tubing to the pillow blocks, lead screw nuts and slides. The oil flow could be restricted by a grub screw with a small orifice, say 0.75 mm. [It's a long time since I drilled such a small hole, the smallest was around 0.5 mm and you break a lot of these tiny drills!]
No I haven 't done this yet!
PS Don't forget to post the info about your CNC conversion!