In the times before it was acceptable to use a belt for keeping your breeches up or braces being invented, this was the way to stop them falling down. Is the 1640s however this had developed into metal eyes sewn to the doublet and matching hooks on the outside of the waistband of the breeches. John Belasyse painted by Gilbert Jackson in 1636 has no obvious external ties joining his doublet and breeches.
At first sight, from the perspective of the 21st century this seems to be one of those stupid fashions that flies in the face of practicality. At Shakespeare's Globe for instance it quickly became apparent in their original practice productions that fast costume changes are not possible in a properly laced doublet and breeches. A large pair of scissors was their solution. Photo on the right shows a detail from a reproduction doublet from the early part of the century. However joining the two had been used for centuries and hooks and eyes were used for a long period in the 17th century, up until the fashionable doublet became so short that there was a gap between it and the breeches to show off the shirt ruffles. Such a long tradition must indicate that this is a practical way to wear clothes.
I'm not sure however how far this practice was carried out during the Civil War for those serving in the armies. There is no mention in any records of hooks and eyes being purchased for soldier's suits, whereas things like buttons are listed frequently. It would also be tricky if you had your own breeches to get an issued coat with eyes that matched the hooks. who knows, as there is no evidence either way we can't tell. However as I have related, it was something that a lot of men were used to, so maybe they went out of their way to customize their clothes. It would make a kind of sense.