First of all go and download the technical manual from digitalwatchlibrary or one of the Seiko sites. The 0624 is very simple inside and any repairable faults are usually done by cleaning/careful reassembly. The fault you describe sounds like a poor earth connection to the LCD panel. Inside the 0624 you have the LCD panel sitting directly over the main DIL MOS chip lying on its back with its legs in the air (stop making your own jokes!). The pins on the chip make direct contact with the panel and must be very carefully adjusted to all be at the same height (detailed in tech manual and something of a black art). Be aware if you remove the chip and rear frame from the plastic LCD carrier getting the two back together without damaging the chip pins is..... fun! I use two strips of 100g white paper cut to the cutout size and place these between the pins and carrier to act as guides and to prevent the pins getting bent (need photo!), when the two parts are back together you pull the two bits of paper out and all done with no bad language!
It doesn't sound like a button spring contact problem but always worth inspecting/cleaning/lubricating these (microscopic drop of red DeoxIT on Seiko GP contacts works wonders).