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<apex:page controller="testctrl" docType="html-5.0">
<apex:form >
  <apex:pageBlock id="myPgBlk">
      <apex:pageBlockSection title="My Pageblocksection" columns="1">
          <apex:inputCheckbox label="Choose a custom date" value="{!chk}">
              <apex:actionSupport event="onchange" rerender="myPgBlk" action="{!NULL}"/>
          </apex:inputCheckbox>
          <apex:input label="Cust Date" value="{!custDate}" type="date" rendered="{!chk == true}"/>
      </apex:pageblockSection>
  </apex:pageBlock>
</apex:form>
</apex:page>


public class testctrl {Load}
    public Boolean chk {get; set;}
    public Date custDate {get; set;}
    public testctrl() {
        chk = false;
        custDate = Date.today();
    }
}