Try this...

have two diff pageblock sections and render it on a certain value(for ex : a boolean value which is der in de controller)

in the controller have two diff methods for right and left buttom clicks.

Right sets the boolean value to false and left sets the boolean value to true and render the two pageblock sections.

<apex:page standardController="Account" tabStyle="account" extensions="thai">
    <apex:form >
        <apex:pageBlock >
            <apex:pageBlockButtons >
                <apex:commandButton value="Right" action="{!getright}"/>
                <apex:commandButton value="Left" action="{!getleft}"/>
            </apex:pageBlockButtons>
            <apex:pageBlockSection id="right" rendered="{!IF(tri=false, true, false)}">
            <apex:dataTable value="{!account}" var="a">
            <apex:column >
            <apex:inputCheckbox value="{!a.hai__c}"/>
            </apex:column>
            <apex:column >
            <apex:facet name="header">Name</apex:facet>
            {!a.name}
            </apex:column>
            </apex:dataTable>
            </apex:pageBlockSection>
            
            <apex:pageBlockSection id="left" rendered="{!IF(tri=true, true, false)}">
            <apex:dataTable value="{!account}" var="a">
             <apex:column >
            <apex:facet name="header">Name</apex:facet>
            {!a.name}
            </apex:column>
            <apex:column >
            <apex:inputCheckbox value="{!a.hai__c}"/>
            </apex:column>
            </apex:dataTable>
            </apex:pageBlockSection>
        </apex:pageBlock>
    </apex:form>
</apex:page>

Extension:

public class thai{

      public thai(ApexPages.StandardController controller) {

    }


Boolean tri = false;

public Boolean gettri(){
return tri;
}

public void getleft(){
tri = true;
}
public void getright(){
tri = false;
}

}