public class DateIssueCont {
@AuraEnabled
public static void saveRecord( Custom_Object__c rec ){
    insert rec;
}
}

<aura:application controller="DateIssueCont">
<aura:attribute name="record" type="Custom_Object__c" default="{'sobjectType': 'Custom_Object__c', 'Date__c': ''}" />
<ui:inputDate value="{!v.record.Date__c}" displayDatePicker="true" />
<aura:handler name="init" value="{!this}" action="{!c.init}"/>
<div aura:id="errors" class="slds-hide">
    <div class="slds-form-element" >
        <div class="slds-form-element__control">
            <ui:message title="Error" severity="error" >
                <ui:outputText aura:id="ErrorMessage" value=""/>
            </ui:message>
        </div>
    </div>
</div>
<button class="slds-button slds-button--neutral slds-button--brand" onclick="{!c.perform}">Perform Business Logic</button>
</aura:application>

({
init : function(component, event, helper) {
    component.set("v.record.Date__c", '2016-1-1' );
},

perform : function(component, event, helper) {
    var rec = component.get( 'v.record' );
    var saveRecord = component.get( "c.saveRecord" );
    saveRecord.setParam( "rec", rec );
    saveRecord.setCallback(this, function(response) {
        var state = response.getState();
        if (component.isValid() && state === "SUCCESS") {
        } else if (state === "ERROR"){
            var errors = response.getError();
            if (errors && errors[0] && errors[0].message) {
                $A.util.removeClass( component.find("errors"), "slds-hide");
                component.find("ErrorMessage").set("v.value", errors[0].message);
            }
            if (errors && errors[0] && errors[0].pageErrors && errors[0].pageErrors[0] && errors[0].pageErrors[0].message ) {
                $A.util.removeClass( component.find("errors"), "slds-hide");
                component.find("ErrorMessage").set("v.value", errors[0].pageErrors[0].message);
            }
            if (errors && errors[0] && errors[0].fieldErrors  ) {
                $A.util.removeClass( component.find("errors"), "slds-hide");
                for ( var u in errors[0].fieldErrors) {
                    component.find("ErrorMessage").set("v.value", errors[0].fieldErrors[u][0].message);
                }

            }
        }
    });
    $A.enqueueAction(saveRecord);
}
})

component.set("v.record.Date__c", '2016-1-1' );

component.set("v.record.Date__c", '2016-01-01' );

rec.Date__c = new Date( rec.Date__c );

var rec = component.get( 'v.record' );