There were a few errors. Here is some updated code and a description of what was going on:

React App.js:

import React, { Component } from 'react';
import axios from 'axios';

class Create extends Component {
  constructor(props) {
    super(props);

    this.state = {
      bookID: '',
      bookTitle: '',
      bookAuthor: '',
    };
  }

  handleInputChange = e => {
    this.setState({
      [e.target.name]: e.target.value,
    });
  };

  handleSubmit = e => {
    e.preventDefault();

    const { bookID, bookTitle, bookAuthor } = this.state;

    const book = {
      bookID,
      bookTitle,
      bookAuthor,
    };

    axios
      .post('http://localhost:3001/create', book)
      .then(() => console.log('Book Created'))
      .catch(err => {
        console.error(err);
      });
  };

  render() {
    return (
      <div>
        <br />
        <div className="container">
          <form onSubmit={this.handleSubmit}>
            <div style={{ width: '30%' }} className="form-group">
              <input
                type="text"
                className="form-control"
                name="bookID"
                placeholder="Book ID"
                onChange={this.handleInputChange}
              />
            </div>
            <br />
            <div style={{ width: '30%' }} className="form-group">
              <input
                type="text"
                className="form-control"
                name="bookTitle"
                placeholder="Book Title"
                onChange={this.handleInputChange}
              />
            </div>
            <br />
            <div style={{ width: '30%' }} className="form-group">
              <input
                type="text"
                className="form-control"
                name="bookAuthor"
                placeholder="Book Author"
                onChange={this.handleInputChange}
              />
            </div>
            <br />
            <div style={{ width: '30%' }}>
              <button className="btn btn-success" type="submit">
                Create
              </button>
            </div>
          </form>
        </div>
      </div>
    );
  }
}

export default Create;

1. You were getting errors for using class and not className. class is a reserved word in react.js and should not be used.
2. You were using the default post method which I would not recommend. I split the post out into it's own action and used the common library axios to make the CORS post call. I also created a function to handle the input changing on every key press with react.js.
3. I added state to your component. This is common when there are form inputs to store them in state. I also changed the name of your variables to be title case which is the common way to write code variables.

Node.js index.js:

const express = require('express');
const logger = require('morgan');
const cors = require('cors');

const app = express();

//use cors to allow cross origin resource sharing
app.use(
  cors({
    origin: 'http://localhost:3000',
    credentials: true,
  })
);

app.use(logger('dev'));
app.use(express.json());
app.use(express.urlencoded({ extended: false }));

let books = [];

app.get('/home', function(req, res) {
  console.log('Inside Home Login');
  res.writeHead(200, {
    'Content-Type': 'application/json',
  });
  console.log('Books : ', JSON.stringify(books));
  res.end(JSON.stringify(books));
});

app.post('/create', function(req, res) {
  const newBook = {
    BookID: req.body.bookID,
    Title: req.body.bookTitle,
    Author: req.body.bookAuthor,
  };

  books.push(newBook);
  console.log(books);
});

//start your server on port 3001
app.listen(3001, () => {
  console.log('Server Listening on port 3001');
});

1. You weren't parsing the body of the req, so it was coming back as undefined. I added app.use(express.json()); and app.use(express.urlencoded({ extended: false })); which should take care of most of the issues.
2. I updated the req.body variables to match those coming over from React.
3. I added the module morgen which you see here app.use(logger('dev')); this is helpful by showing all your requests and statuses for dev purposes. In this case, it was showing that you were getting a 500 (internal server error) because express couldn't read bookID of undefined (because the body wasn't being parsed).

This should be working now, let me know if you have any problems.