How do I register a Javascript variable on the server side (backend) and access it on the client side (Javascript file), without a hidden field, Literal, etc.?
You can use the RegisterClientScriptBlock-Function from the Page's ClientScriptManager.
Page.ClientScript.RegisterClientScriptBlock(Page.GetType, "initMyClientVariable", "var myClientVariable=null;", True)
EDIT: according to your new informations, that you want to register a client array, use ClientScriptManager's RegisterArrayDeclaration Method.
VB.Net example:
Dim myArrayValue As String = """1"", ""2"", ""text"""
Page.ClientScript.RegisterArrayDeclaration("myClientArray", myArrayValue)
According to the new information in my comments that you need access to that variable from an external js-file: you should pass the js-array as argument to the function in the js-file. For example:
callFunctionInJsFile(checkBoxes);