There are two things you need to change in your code. Make your startDate and endDate as Date data type.
public class Unavailable_Dates_Controller
{
public Date startDate{get;set;}
public Date endDate{get;set;}
}
As per document apex:inputField can only be use with sObject fields.
An HTML input element for a value that corresponds to a field on a Salesforce object.
There are some related post i found.
Lets explore all the ways to form Date pickers
a)Using inputfield is best solution .
Disadvantage:Field level security and profile settings .If your object dont have edit access or field is hidden for any profile the same will apply and there is no way to bypass this security if you wish to by pass for various profiles (Note this is also advantage if your requirement is not to allow edit based on profile)
b)Hackish solution with input text
Visualforce page:
<apex:page controller="datePicker" id="mypage">
<apex:form>
Date: <apex:inputText value="{!datename}" size="10" id="demo" onfocus="DatePicker.pickDate(false, this , false);" />
</apex:form>
</apex:page>
Apex Controller:
public class DatePicker {
public String datename {get; set;}
}
Disadvantage :hacked the JavaScript of salesforce and not officially recommended .
c)New trend after winter 14
<apex:page docType="html-5.0" />
<apex:pageBlockSection >
<apex:outputLabel ></apex:outputLabel>
<apex:input label="datePicker" value="{! fDate }" type="auto"/>
</apex:pageBlockSection>
</apex:page>
public class winter14Ctrl {
public Date fDate { get; set; }
}
Advantage :This is native to salesforce and safely recommended
Disadvantages: Not supported in Firefox and IE. Source: http://caniuse.com/#search=input%20type Your all the instances must be updated to winter 13
d)Using jquery component to do this .Jquery library or jquery UI will have this component
disadvantage:More developer skills needed and lot of testing with different browser
Source : link
<apex:inputField />
works only for sObject fields AFAIK. You can't bind class fields there and get Datepicker works