First I'll answer your question on the surface and then I'll drill down and look at the bigger picture.
Listboxes are unique controls in that they allow you to select one or multiple items, but the option to select multiple is arcane and unfamiliar to most users. This makes using them for general purposes discouraged at best and a serious usability problem at worst.
Instead of using a list box here I would put the assigned assets in a table with one asset in each row and an associated start and end date control in columns beside it. This is much more obvious: you separate the process of assigning assets from selecting dates for assigned assets, and you make the list more scannable as well as reviewable (since you can look through the list to verify your selected dates for each asset).
Who else does start and end date controls well?
The travel industry has to design user interfaces that solve this problem all the time. One leading site here is Hipmunk. Let's see what they came up with:
They allow you to input a date in plain English and then below they display a calendar to give you feedback, explaining that the system understood what you meant. This is a great approach and one that I recommend you reproduce for maximum awesome points.
Looking at your screenshot is making my brain explode. My first scan of the controls doesn't inform me of how this user interface is supposed to work. After looking at it for a while I can figure it out, but that's as an experienced designer. What about the people expected to use this?
I would strongly recommend redesigning this part of your UI. Let's walk through an example. Let's say I'm a photographer and I want to check out a camera to use for a shoot. I want to be able to find "Photography" as a service, then "Cameras" as the type of equipment, then "Nikon D5000" as the actual camera. Once selected I want to tell the site when I need it.
The problem is that instead of allowing me to just find the camera directly, you're forcing me to map my mental model to your domain structure. This is inconvenient for me. A better approach would be to just let me type in what I'm looking for, and you figure out how to find it for me.
For instance, a more human approach here would be to offer me a search field with suggest. When I type "Nikon", you suggest "Nikon 5000D (Photography > Cameras)". When I select it, it gets added to a list of assets similar to the one I showed above.