Javascript button visibility depending on a variable

Command Button in salesforce comes with an attribute rendered .You will have to make use of same .

Also to write conditions inside render tag you can use FORMULAE

You can call an action function onclick of the checkbox and process the logic inside your actionfunction and return the boolean and use the boolean on the rendered attribute of the command button .

PSEUDO CODE

<apex:commandButton id="ApproveButton" value="Register and Accept Invitation" action="{!loginAndAccept}" rendered="{!registerBoolean}"/>

Apex:

public boolean registerBoolean {get;set;}
constructor(){
  registerBoolean = true;
}
public pagereference loginAndAccept(){
  registerBoolean = false;
  return true;
}

Try this code using RemoteAction :

VF page :

<apex:inputCheckbox onchange="changeState(this)" rerender="buttonsDiv"/>

<div id="buttonsDiv">
    <apex:commandButton id="ApproveButton" value="Register and Accept Invitation" action="{!loginAndAccept}" rendered="{!hasUser== 'false'}"/>
    <apex:commandButton id="LoginButton" value="Login and Accept Invitation" action="{!registerAndAccept}" rendered="{!hasUser == 'true'}"/>
</div>

<script type="text/javascript">    
    function changeState(state) {
        var isChecked = (state.checked) ? 'false' : 'true';

        Visualforce.remoting.Manager.invokeAction(
            '{!$RemoteAction.YourController.changeChkState}',
            isChecked,
            function (result, event) {
            }, 
            {escape: false}
        );
    }
</script>

Controller :

public class YourController {
    public String hasUser {get; set;}

    @RemoteAction
    public void changeChkState(state) {
       hasUser = state;
    }
}