On my laptop I have a well secured Linux installation that I would like to stay this way. It has in the past and could in the future contain confidential medical data I use for my study. I also have a Windows installation I use for more casual usage (games, downloading stuff ext.). Is it possible to hack my linux installation via the windows partition? This goes both for directly accessing the data and installing a program somewhere where Linux might find it and run it.
If you want to stay safe you can just ensure your Linux home and root partition are encrypted and use UEFI to ensure your Linux kernel isn't modified. Then it will be impossible for your Windows box to touch the rest of the OS. A less elegant (and insufficient from a legal standpoint if you need to guarantee the data is well protected) but maybe easier approach is to use a cryptic filesystem format that Windows doesn't support but Linux does to store your data :-)
Note, as pointed out by Polynomial in the comments, that cryptsetup (until 2013) used CBC as a default method for encryption which opens up for malleability attacks, leading to the ability to make predictable changes to specific locations on the hard drive's content. Make sure to use EBC instead and remember that you cannot guarantee integrity on your system.
This being said, I would be seriously more worried about my Linux box than about cross-OS attacks if I were you. How do you currently guarantee a rogue or compromised Linux app isn't accessing your study participant data?
A dual boot just adds a new attack vector: when the alternate OS is running, if can be remotely attacked. The attacker could then use the alternate OS to read (complex in general case, very complex if encryption is used) modify (slightly more hard than reading) or destroy (trivial if admin priviledges are present(*)) the main OS. But you should remember that a dual boot only slightly eases local attacks. If attackers have physical access to the machine, they can boot from a removable media (DVD or memory stick) which gives them admin priviledges on this new OS. Or if no DVD reader nor USB slot is present take the disk out of the case.
(*) In any OS I know, admin priviledges are both required and enough to access any disk at the record level. Writing random data will easily destroy all previous informations from any disk...
Dual Boot can (but doesn't have to) hurt security. It just means you have another attack vector to protect against.
The biggest concern (tied to dual-boot) is that someone could use some level of compromise on one side to cause further harm on the other side. For example, if someone managed to get arbitrary code running on the windows side and the linux file system was mounted rw, they could modify scripts in /etc/init.d/ to do arbitrary damage as root. This would be a full and persistent exploit of the linux side upon its next boot.
Remember, the booted OS essentially as access to the non-booted OS as a data drive (similar to a drive that has been pulled and put in to a second computer). The same mechanisms to protect against that protect in a dual-boot scenario. Disk encryption is highly recommended. If you are protecting medical data, you probably should be doing that anyway. Steve's answer above is quite good covering the encryption options, so I won't repeat them here.