Yes you can do that. You have to update the Salesforce generated code as below. Only one issue user has to disable pop-up blocker.
Salesforce generated code.
<html>
<head><title>AskNow LiveAgent Chat</title></head>
<body>
<div id="chat-icon">
<img id="liveagent_button_online_57328000000L25v" style="display: none; border: 0px none; cursor: pointer" onclick="liveagent.startChat('57328000000L25v')" src="https://ap2.salesforce.com/resource/1455000961000/OnlineChatButton" />
<img id="liveagent_button_offline_57328000000L25v" style="display: none; border: 0px none; " src="https://ap2.salesforce.com/resource/1455001009000/OfflineChatButton"/>
</div>
<script type='text/javascript' src='https://c.la10.salesforceliveagent.com/content/g/js/36.0/deployment.js'></script>
<script type="text/javascript">
function setName() {
document.getElementById("prechat_field").value = "Hellloooo";
return true;
}
liveagent.init('https://d.la10.salesforceliveagent.com/chat', '57228000000GpYe', '00D28000001KSvw');
if (!window._laq) { window._laq = []; }
window._laq.push(function(){
liveagent.enableLogging();
liveagent.setChatWindowHeight('720');
liveagent.showWhenOnline('57328000000L25v', document.getElementById('liveagent_button_online_57328000000L25v'));
liveagent.showWhenOffline('57328000000L25v', document.getElementById('liveagent_button_offline_57328000000L25v'));
});
</script>
</body>
</html>
Updated code
<html>
<head><title>AskNow LiveAgent Chat</title></head>
<body>
<div id="chat-icon">
<form>
<!-- Detail inputs -->
<p>First Name:</p><input type="text" id="firstName" /><br />
<p>Last Name:</p><input type="text" id="lastName" /><br />
<p>Email:</p><input type="text" id="email" /><br />
<input type="button" value="Submit" onClick="startChat();return false;">
</form>
</div>
<script type='text/javascript' src='https://c.la10.salesforceliveagent.com/content/g/js/36.0/deployment.js'></script>
<script type="text/javascript">
function setName() {
document.getElementById("prechat_field").value = "Hellloooo";
return true;
}
function startChat(){
liveagent.init('https://d.la10.salesforceliveagent.com/chat', '57228000000GpYe', '00D28000001KSvw');
if (!window._laq) { window._laq = []; }
window._laq.push(function(){
liveagent.enableLogging();
liveagent.setChatWindowHeight('720');
liveagent.showWhenOnline('57328000000L25v', document.getElementById('liveagent_button_online_57328000000L25v'));
liveagent.showWhenOffline('57328000000L25v', document.getElementById('liveagent_button_offline_57328000000L25v'));
liveagent.addCustomDetail('First Name', document.getElementById('firstName').value);
liveagent.addCustomDetail('Last Name', document.getElementById('lastName').value);
liveagent.addCustomDetail('Email', document.getElementById('email').value);
liveagent.addButtonEventHandler('57328000000L25v',function (e){
if(e == liveagent.BUTTON_EVENT.BUTTON_AVAILABLE){
liveagent.startChat('57328000000L25v');
}
});
});
}
</script>
</body>
</html>