This article outlines how to persist the file shares across VM reboots - https://azure.microsoft.com/en-us/documentation/articles/storage-dotnet-how-to-use-files/
I copied the relevant snippet below.
Persist your storage account credentials for the virtual machine
Before mounting to the file share, first persist your storage account credentials on the virtual machine. This step allows Windows to automatically reconnect to the file share when the virtual machine reboots. To persist your account credentials, run the cmdkey command from the PowerShell window on the virtual machine. Replace <storage-account-name> with the name of your storage account, and <storage-account-key> with your storage account key.
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cmdkey /add:<storage-account-name>.file.core.windows.net /user:<storage-account-name> /pass:<storage-account-key>
Windows will now reconnect to your file share when the virtual machine reboots. You can verify that the share has been reconnected by running the net use command from a PowerShell window.
Note that credentials are persisted only in the context in which cmdkey runs. If you are developing an application that runs as a service, you will need to persist your credentials in that context as well.
Once you have a remote connection to the virtual machine, you can run the net use command to mount the file share, using the following syntax. Replace <storage-account-name> with the name of your storage account, and <share-name> with the name of your File storage share.
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net use <drive-letter>: \\<storage-account-name>.file.core.windows.net\<share-name>
example :
net use z: \\samples.file.core.windows.net\logs
Since you persisted your storage account credentials in the previous step, you do not need to provide them with the net use command. If you have not already persisted your credentials, then include them as a parameter passed to the net use command, as shown in the following example.
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net use <drive-letter>: \\<storage-account-name>.file.core.windows.net\<share-name> /u:<storage-account-name> <storage-account-key>
example :
net use z: \\samples.file.core.windows.net\logs /u:samples <storage-account-key>
You can now work with the File Storage share from the virtual machine as you would with any other drive. You can issue standard file commands from the command prompt, or view the mounted share and its contents from File Explorer. You can also run code within the virtual machine that accesses the file share using standard Windows file I/O APIs, such as those provided by the System.IO namespaces