2 added 1 character in body
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You should use && instead of & for joining two logical conditions.
So, your code should look like something this:

List<Profile> PROFILE = [SELECT Id, Name FROM Profile WHERE Id=:userinfo.getProfileId() LIMIT 1];
Public String MyProflieName = PROFILE[0].Name;
List<User> Use = new List<User>([SELECT Phone, Id FROM User WHERE Id = : UserInfo.getUserId()]);
for(Case c:Trigger.New)
    {
        for(User u: Use)
        {
            System.Debug(
            (c.RecordType.Name='Client Service')
            &&((c.Categories__c !='DCS Client Services')||(c.Categories__c !='DCS Technical Support'))
            &&((c.Account_Segmentation__c != 'Z2')||(c.Account_Segmentation__c != 'Z3')||(c.Account_Segmentation__c != 'Z4')||(c.Account_Segmentation__c !=' Z5')||(c.Account_Segmentation__c != 'D1'))
            &&((MyProflieName=='DCS Client Services')||(MyProflieName=='DCS Technical Support'))
            &&((c.Origin='Email')||(c.Origin='Phone')||(c.Origin='Chat')||(c.Origin='Internal'))
            &&(!((u.FirstName='PBI')&&&(u.LastName='Default')))
            );

        }
    }
}

You should use && instead of & for joining two logical conditions.
So, your code should look like something this:

List<Profile> PROFILE = [SELECT Id, Name FROM Profile WHERE Id=:userinfo.getProfileId() LIMIT 1];
Public String MyProflieName = PROFILE[0].Name;
List<User> Use = new List<User>([SELECT Phone, Id FROM User WHERE Id = : UserInfo.getUserId()]);
for(Case c:Trigger.New)
    {
        for(User u: Use)
        {
            System.Debug(
            (c.RecordType.Name='Client Service')
            &&((c.Categories__c !='DCS Client Services')||(c.Categories__c !='DCS Technical Support'))
            &&((c.Account_Segmentation__c != 'Z2')||(c.Account_Segmentation__c != 'Z3')||(c.Account_Segmentation__c != 'Z4')||(c.Account_Segmentation__c !=' Z5')||(c.Account_Segmentation__c != 'D1'))
            &&((MyProflieName=='DCS Client Services')||(MyProflieName=='DCS Technical Support'))
            &&((c.Origin='Email')||(c.Origin='Phone')||(c.Origin='Chat')||(c.Origin='Internal'))
            &&(!((u.FirstName='PBI')&(u.LastName='Default')))
            );

        }
    }
}

You should use && instead of & for joining two logical conditions.
So, your code should look like something this:

List<Profile> PROFILE = [SELECT Id, Name FROM Profile WHERE Id=:userinfo.getProfileId() LIMIT 1];
Public String MyProflieName = PROFILE[0].Name;
List<User> Use = new List<User>([SELECT Phone, Id FROM User WHERE Id = : UserInfo.getUserId()]);
for(Case c:Trigger.New)
    {
        for(User u: Use)
        {
            System.Debug(
            (c.RecordType.Name='Client Service')
            &&((c.Categories__c !='DCS Client Services')||(c.Categories__c !='DCS Technical Support'))
            &&((c.Account_Segmentation__c != 'Z2')||(c.Account_Segmentation__c != 'Z3')||(c.Account_Segmentation__c != 'Z4')||(c.Account_Segmentation__c !=' Z5')||(c.Account_Segmentation__c != 'D1'))
            &&((MyProflieName=='DCS Client Services')||(MyProflieName=='DCS Technical Support'))
            &&((c.Origin='Email')||(c.Origin='Phone')||(c.Origin='Chat')||(c.Origin='Internal'))
            &&(!((u.FirstName='PBI')&&(u.LastName='Default')))
            );

        }
    }
}
1
source | link

You should use && instead of & for joining two logical conditions.
So, your code should look like something this:

List<Profile> PROFILE = [SELECT Id, Name FROM Profile WHERE Id=:userinfo.getProfileId() LIMIT 1];
Public String MyProflieName = PROFILE[0].Name;
List<User> Use = new List<User>([SELECT Phone, Id FROM User WHERE Id = : UserInfo.getUserId()]);
for(Case c:Trigger.New)
    {
        for(User u: Use)
        {
            System.Debug(
            (c.RecordType.Name='Client Service')
            &&((c.Categories__c !='DCS Client Services')||(c.Categories__c !='DCS Technical Support'))
            &&((c.Account_Segmentation__c != 'Z2')||(c.Account_Segmentation__c != 'Z3')||(c.Account_Segmentation__c != 'Z4')||(c.Account_Segmentation__c !=' Z5')||(c.Account_Segmentation__c != 'D1'))
            &&((MyProflieName=='DCS Client Services')||(MyProflieName=='DCS Technical Support'))
            &&((c.Origin='Email')||(c.Origin='Phone')||(c.Origin='Chat')||(c.Origin='Internal'))
            &&(!((u.FirstName='PBI')&(u.LastName='Default')))
            );

        }
    }
}