Most common silver or shiny screws and bolts are either zinc plated or made of zinc or zinc alloy. These are usually the most economical option sold by hardware stores. That looks to be the type of screws you have used. The zinc coating is somewhat corrosion resistant and that is one of the reasons it is used. If that's the case, your tools should be totaly fine as long as the storage board is in a fairly dry place. I'd argue that the MDF or masonite board will degrade (it absobrbs moisture like a sponge) before the screws have enough chance to rust.
Other common screws are black, this is a "black oxide" coating. In my experience this ones are more prone to corrosion and I'd avoid them for this use.
Some screws and bolts are actually sold coated in grease or oil (These are the actual cheapest option in my usual supply store) and they are quick to rust, they are low grade steel or mild steel, once you clean them of the oil, they look dark, dull gray. Those I would also avoid for this project.
However, there are some other suggestions you may find useful:
Coat the screws with heat-shrink tubing. This is an isolation material for electrical connections and components, it contracts when heated. I have used a hairdrier to shrink it, but a lighter, soldering iron or a hor air gun can also be used.
If you have around a lenght of small diameter hose, you can use it around the screws also. I hapen to have a hose from an air compressor, It's about inner 1/4" diameter, and about 1/2" outer. Cut it in short "sleeves", unscrew, slide the screw into the sleeve and install it again. Another source of hose, a bit thinner, can be the hoses from nebulizer or similar medical equipment, those vary around 1/8" diameter and won't be so thick, alowing for tools with smaller hanging holes.
Substitute the screws for dowels. You can make dowels from used chopsticks, or bamboo skewers, etc.