Need to install the same version of SQL Express(Example: 2012) you used for development in client machines. By default sql express will be installed as named instance with name ".\sqlexpress" so your connection string need to point to the server as ".\sqlexpress" or "(local)\sqlexpress". You can change the named instance during installation if needed. Also users can add themself as SQL admin to sql express during installation so if they do that then your application should work fine.
This link shows step by step installation of sql express 2012